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10   SPACES WITH OPERATORS. ORBIT SPACES       51

x e E and all, s, t in G. Everything we have said can be immediately transposed
to this situation; the set of orbits is sometimes denoted by G\E. For example,
a subgroup H of a topological group G acts continuously on the right on G by
the action (s, x)\~+xs.

PROBLEMS

1.    Let G be a locally compact metrizable group and E a locally compact metrizable space
on which G acts continuously. For each pair of subsets K, L in E let P(K, L) denote
the set of all s e G such that (s  K) n L ^ 0.

(a)    Show that if K is compact and L is closed in G, then the set P(K, L) is closed
in G. The group G is said to act properly on E if P(K, L) is a compact subset of G
whenever K and L are compact subsets of E. (This will always be the case if G is
compact.)

(b)    Show that, if G acts properly on E, then F  K is closed in E whenever F is closed
in G and K is a compact subset of E. In particular, for each x e E, the orbit G  x is
closed in E.

(c)    Under the same hypotheses, for each x e E the stabilizer S* of x is a compact
subgroup of G, and the canonical map G/S* -> G  x is a homeomorphism.

(d)    Under the same hypotheses, the orbit space E/G is Hausdorff.

2.    For each pair (a, t) of real numbers such that a^l, the point fa(t) eR2 is defined as
follows:

if   /<~4r.

fl-f   1)

if   -

a-f 1

' )

+ I/

if

Let Cfl denote the set of points fa(t) (t e R) and let E be the union of the sets Ca (a ^ 1)
and the lines D', D*, where D' (resp. D*) is the set of points (/, 1) (resp. (t, 1)) with
/eR.

The additive group R acts on the locally compact space E as follows:
(1) s - (t, -1) = (s 4- 1, -1); (2) 5 /(/) =/0 + /) for a  1 ; (3) s - (t, 1) * (/- j, 1).
Show that R acts continuously and freely on E, that the orbits R  z are closed sets and
that the canonical mapping R~>R  z is a homeomorphism, but that the orbit space
E/R is not Hausdorff.

3.   The group Z acts continuously and properly on R2 as follows :

The orbit space M (the Mobius strip) is metrizable and locally compact. If TT : R2 - M
is the canonical mapping, the restriction of TT to E  D" (notation of Problem 2) is
injective, and ?r(E  DO = 7r(E) is locally compact. The group R acts on u(E  DO
by the law s - rr(z) = TT($  z) for all z e E  D*. Show that R acts continuously andgroup G on a set E is a mapping (s, x)\-*x- s